J. Korean Math. Soc. 40 (2003), No. 6, pp. 1075–1083
CURVED DOMAIN APPROXIMATION IN DIRICHLET’S PROBLEM Miyoung Lee, Sang Mok Choo, and Sang Kwon Chung Abstract. The purpose of this paper is to investigate the piecewise polynomial approximation for the curved boundary. We analyze the error of an approximated solution due to this approximation and then compare the approximation errors for the cases of polygonal and piecewise polynomial approximations for the curved boundary. Based on the results of analysis, p-version numerical methods for solving Dirichlet’s problems are applied to any smooth curved domain.
1. Introduction We consider the Poisson equation (1.1)
−∆u = F
in Ω,
with the Dirichlet boundary condition u=0
on ∂Ω,
where F ∈ C ∞ (Ω). Here Ω is a bounded domain in the plane with an infinitely differentiable curved boundary ∂Ω which may be neither polygonal nor polynomial. It is usually assumed that the boundary ∂Ω of Ω is either polygonal structures or polynomials when some numerical methods are implemented and corresponding numerical estimations are obtained(see [1, 2, 3, 4]). Thus if a curved boundary is neither polygonal nor polynomial, these results may not be held. Received February 28, 2003. 2000 Mathematics Subject Classification: Primary 65M06, 65M12, 65M15. Key words and phrases: curved domain, polynomial boundary approximation, p-version, Poisson equation.
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Strang and Berger [5] and Thom´ee [6] obtained the error estimates for the following equation (1.2)
−∆uh = F
in Ωh ,
uh = 0
on ∂Ωh ,
where a polygonal domain Ωh approximates a curved domain Ω and h is the maximum length of the edges of ∂Ωh . Due to these theoretical results, the implementation of the numerical h-version method to a curved domain is meaningful. To obtain more accurate results for the numerical solution on domains with curved boundary, we should apply a p-version method to the curved boundary. But we are aware of no studies that have examined p-version boundary approximation. The purpose of this paper is to generalize the polygonal boundary approximation using polynomial approximation. We will approximate the curved boundary by a piecewise polynomial boundary and analyze the errors of the approximated solution due to this polynomial boundary approximation. In section 2, the error estimates in the maximum norm are discussed. In section 3, the L2 -error estimates for gradients of numerical solutions are discussed. 2. Error estimates in the maximum-norm We assume that the boundary ∂Ω is a union of subboundaries (fi (s), gi (s)) such that ∂Ω = ∪1≤i≤k {(fi (s), gi (s)) : 0 ≤ s ≤ 1} with (fi (1), gi (1)) = (fi+1 (0), gi+1 (0)), (fk (1), gk (1)) = (f1 (0), g1 (0)), where fi , gi ∈ C ∞ [0, 1]. Let Ωp be the curved domain with boundary ∂Ωp = ∪1≤i≤k {(Πi (s), Ψi (s)) : 0 ≤ s ≤ 1}, where Πi and Ψi be the Lagrange interpolating polynomials of degree pi satisfying Πi (
` ` ) = fi ( ), pi pi
Ψi (
` ` ) = gi ( ), pi pi
` = 0, . . . , pi .
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Then the corresponding polynomial approximation problem for (1.1) becomes (2.1)
−∆up = F
in Ωp ,
up = 0 on ∂Ωp .
Let | · |Ω denote the sup-norm in Ω and k · kΩ denote L2 -norm over Ω, |u|m,Ω = max|α|≤m |Dα u|Ω and |x|2 = |(x1 , x2 )|2 = x21 + x22 . Using the maximum principle, we obtain the following theorem. ˜ where Ω ˜ ) Ω. Then there exists a Theorem 2.1. Let F ∈ C ∞ (Ω) constant C which depends only on Ω such that 1 1 (p +1) |u − up |Ω∩Ωp ≤ C max {|fi i |[0,1] ( )pi +1 , 1≤i≤k pi + 1 pi 1 1 pi +1 (pi +1) |gi |[0,1] ( ) }|F |Ω˜ , pi + 1 pi (pi +1)
where Ω ∩ Ωp = (Ω ∩ Ωp ) ∪ ∂(Ω ∩ Ωp ) and fi derivative of fi .
is the (pi + 1)st
Proof. Let d(x, Ω) be the distance between Ω and x. Let Ω² = ∪1≤i≤k {x : d(x, Ωi ) < ²i }, where ∂Ωi = {(fi (s), gi (s)) : 0 ≤ s ≤ 1}, 1 1 1 1 (p +1) (p +1) ²i = max{|fi i |[0,1] ( )pi +1 , |gi i |[0,1] ( )pi +1 } pi + 1 pi pi + 1 pi ˜ for large p. so that Ω ∪ Ωp ⊂ Ω² ⊂ Ω ² Let u be the solution of −∆u² = F
in Ω² ,
u² = 0
on ∂Ω² .
Note that ∂Ω² ∈ C ∞ except finite number of points. Then for x ∈ ∂Ω ∪ ∂Ωp , there exists yx ∈ ∂Ω² with d(x, ∂Ω² ) = d(x, yx ). Since u² − u is harmonic in Ω, we obtain for some C |u² − u|Ω ≤ |u² − u|∂Ω = sup |u² (yx ) − u² (x)| x∈∂Ω
≤ sup |yx − x||u² |1, x∈∂Ω
≤ max ²i |u² |1, ≤C
Ω² max ²i |∆u² |Ω²
≤ C max ²i |F |Ω˜ .
Ω²
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Miyoung Lee, Sang Mok Choo, and Sang Kwon Chung
On the other hand |up − u² |Ωp ≤ |up − u² |∂Ωp = sup |u² (yx ) − u² (x)| x∈∂Ωp
≤ sup |yx − x||u² |1,Ω² x∈∂Ωp
≤ 2 max ²i |u² |1,Ω² ≤ C max ²i |∆u² |Ω² ≤ C max ²i |F |Ω˜ . Using triangle inequality, we obtain the desired result.
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Remark 2.1. Let {Pj }N 0 be a sequence of points on ∂Ω with P0 = PN and Ωh be a polygonally approximated domain with the maximum length h of edges in a triangulation for Ωh . Thom´ee [5] obtained for convex domain Ω |u − uh |Ωh ≤ Ch2 . Thus the errors of h-version and p-version are |u − uh |Ωh ∩Ωp = O( |u − up |Ωh ∩Ωp = O(
1 ), N2
1 ), (N − 1)N +1
−−−−→ −−−−−−→ respectively, whenever |Pi Pi+1 | = |Pi+1 Pi+2 |, f (q) and g (q) are bounded for q = 0, . . . , N . Therefore we can see that a curved boundary approximation is more accurate than a polygonal boundary approximation. In most numerical studies on a curved domain, the curved boundary is polynomial. In this case, Ω = Ωp for some p. Thus the errors of h-version and p-version are, respectively, |u − uh |Ωh = O( |u − up |Ωh = O(
1 ), N2
1 ). (N − 1)N +1
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3. L2 -error estimates for gradients Without loss of generality, we may assume that the origin belongs to Ω and there exists a positive constant µ such that x · η(x) ≥ µ > 0,
x ∈ ∂Ω,
where η(x) is the outward unit normal vector to the boundary ∂Ω at a point x. For example, a convex domain satisfies the above condition. Let s(x) be the unit tangent vector to the boundary ∂Ω at a point x. Using the Green’s identity, we obtain the following lemma. Lemma 3.1. There exists a constant C such that for a harmonic function v in Ω ∩ Ωp k
∂v ∂v k∂(Ω∩Ωp ) ≤ C| |∂(Ω∩Ωp ) , ∂η ∂s
∂ ∂ where ∂η and ∂s denote differentiations in the direction of normal and tangent vector, respectively.
Proof. Integrating the following identity ¸ 2 · 2 X X ∂v 2 ∂ ∂v ∂v ∂v ∂ {xk ( ) }−2 (xk ) +2 xk ∆v = 0 ∂xk ∂xj ∂xj ∂xj ∂xk ∂xk
j,k=1
k=1
over Ω∩Ωp and using Green’s identity, we obtain for a harmonic function v · ¸ Z 2 X ∂v 2 ∂v ∂v (3.1) x·η ( ) −2 ds = 0, ∂xj ∂n ∂r ∂(Ω∩Ωp ) j=1 where ∂v ∂r denotes the derivative in direction of x = (x1 , x2 ). Note that for unit vectors η = (η1 , η2 ) and s = (−η2 , η1 ), it holds that (3.2)
2 X ∂v 2 ∂v ∂v ( ) = ( )2 + ( )2 . ∂xj ∂η ∂s j=1
Setting x · η = kxkxn ,
x · s = kxkxs ,
∂v ∂v ∂v = (xn + xs )kxk, ∂r ∂η ∂s
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Miyoung Lee, Sang Mok Choo, and Sang Kwon Chung
and using (3.2), we may rewrite (3.1) as · ¸ Z ∂v ∂v ∂v 2 ∂v 2 xn ( ) − xn ( ) − 2xs kxkds = 0. ∂s ∂η ∂s ∂η ∂(Ω∩Ωp ) Thus we obtain for ² > 0 Z ∂v xn ( )2 kxkds ∂η ∂(Ω∩Ωp ) · ¸ Z ∂v 2 ∂v ∂v = xn ( ) − 2xs kxkds (3.3) ∂s ∂s ∂η ∂(Ω∩Ωp ) · ¸ Z 1 ∂v 2 ∂v 2 (xn + 2|xs | )( ) + 2|xs |²( ) kxkds. ≤ ² ∂s ∂η ∂(Ω∩Ωp ) Since there exists a positive constant µ such that x · η ≥ µ > 0,
x ∈ ∂Ω,
and ∂Ωp is any kind of polynomial approximation of ∂Ω, there exists a positive constant λ such that for sufficiently large p x · η ≥ λ > 0,
x ∈ ∂Ωp .
Therefore we obtain x · η ≥ min{µ, λ} > 0,
x ∈ ∂(Ω ∩ Ωp ).
Since there exist constants r0 and r1 such that for all x ∈ ∂(Ω ∩ Ωp ), 0 < r0 ≤ kxk ≤ r1 , we can choose ² in (3.3) such that xn kxk − 2|xs |²kxk > min{µ, λ} − 2r1 ² > 0,
x ∈ ∂(Ω ∩ Ωp ).
It completes the desired proof.
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Using the Green’s formula, we obtain the following theorem. Theorem 3.1. There exists a constant C depending on u such that k∇u − ∇up kΩ∩Ωp (pi +1)
≤ C max {|fi 1≤i≤k
|[0,1] (
3 1 pi + 3 1 (p +1) ) 2 , |gi i |[0,1] ( )pi + 2 }. pi pi
Curved domain approximation in Dirichlet’s problem
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Proof. Since ep = u − up is a harmonic function in Ω ∩ Ωp , using Lemma 3.1, we obtain for some constant C Z ∂ep (3.4) k∇ep k2Ω∩Ωp = ep ds ∂η ∂(Ω∩Ωp ) ∂ep k∂(Ω∩Ωp ) ∂n ∂ep ≤ Ckep k∂(Ω∩Ωp ) | |∂(Ω∩Ωp ) . ∂s ≤ kep k∂(Ω∩Ωp ) k
Note that ∂ep ∂ep ∂ep |∂(Ω∩Ωp ) = | |∂(Ω∩Ωp )∩∂Ω + | |∂(Ω∩Ωp )∩∂Ωp ∂s ∂s ∂s and the second term in the righthand side of (3.5) is ∂ep | |∂(Ω∩Ωp )∩∂Ωp = sup |∇u(xp ) · s(xp )|. ∂s xp ∈∂(Ω∩Ωp )∩∂Ωp (3.5)
|
Let xp = (Πi (s0 ), Ψi (s0 )) and x ˜ = (fi (s0 ), gi (s0 )). Then we obtain (3.6)
|∇u(xp ) · s(xp )| ≤ |∇u(˜ x) · s(xp )| + Ckxp − x ˜k π ˜k} ≤ C{| cos( ± θ)| + kxp − x 2 ≤ C{sin θ + kxp − x ˜k},
where θ is the acute angle between the tangential vectors s(xp ) on ∂Ωp and s(˜ x) on ∂Ω. Using the smoothness of functions fi , gi , Πi , Ψi , the inequality (3.6) becomes |∇u(xp ) · s(xp )| ≤ C{sin θ + kxp − x ˜k} q ≤ C{ (fi0 (s0 ) − Π0i (s0 ))2 + (gi0 (s0 ) − Ψ0i (s0 ))2 + kxp − x ˜k}. Thus we obtain (3.7) ∂ep | |∂(Ω∩Ωp )∩∂Ωp ∂s ·
1 1 pi +1 (p +1) ) , |gi i |[0,1] ( )pi +1 } 1≤i≤k pi pi ¸ 1 1 1 1 (p +1) (p +1) ( )pi +1 , |gi i |[0,1] ( )pi +1 } + max {|fi i |[0,1] 1≤i≤k pi + 1 pi pi + 1 pi 1 1 (p +1) (p +1) ≤ C max {|fi i |[0,1] ( )pi +1 , |gi i |[0,1] ( )pi +1 }. 1≤i≤k pi pi
≤C
(pi +1)
max {|fi
|[0,1] (
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Miyoung Lee, Sang Mok Choo, and Sang Kwon Chung ∂e
Similarly we obtain the inequality (3.7) for | ∂sp |∂(Ω∩Ωp )∩∂Ω . Applying (3.7) and Theorem 2.1 to (3.4), we obtain the desired results. ¤ Remark 3.1. Thom´ee [5] obtained for a convex domain Ω 3
|∇u − ∇uh |Ωh ≤ Ch 2 , where h is the maximum length of the edges in a triangulation for Ωh . Thus we obtain similar results as in Remark 2.1. 4. Concluding remark We approximated a curved boundary by polynomials of variable degrees and obtained the corresponding error estimates. Since a piecewise polynomial boundary approximation for a curved boundary is more accurate than that a polygonal boundary approximation, we were able to show that the numerical solution by p-version methods is more accurate than that obtained by using the usual boundary approximation for a curved boundary. Acknowledgment. The authors are indebted to V. Thom´ee for encouraging and constant interest during the investigation. Also we thank to F. Milner for his stimulating encouragement to realize the value of this study. References [1] M. R. Dorr, The approximation theory for the p-version of the finite element method, SIAM J. Numer. Anal. 21 (1984), no. 6, 1180–1207. [2] M. Lee and F. Milner, Mixed finite element methods for nonlinear elliptic problems: The p-Version, Numer. Methods Partial Differential Equations 12 (1996), 729–741. [3] J. T. Oden and L. Demkowicz, Mixed finite element methods for nonlinear elliptic problems: The p-Version, Comput. Methods Appl. Math. 89 (1991), 11–40. [4] M. Suri, On the stability and convergence of higher order mixed finite element methods for second order elliptic problems, Math. Comp. 54 (1990), 1–19. [5] G. Strang and A. E. Berger, The Change in solution due to change in domain, Proceedings of the AMS summer institute on partial differential equations at Berkeley (1971), 199–205. [6] V. Thom´ ee, Polygonal domain approximation in Dirichlet’s problem, J. Inst. Math. Appl. 11 (1973), 33–44.
Curved domain approximation in Dirichlet’s problem
Miyoung Lee Department of Management Information System Konkuk University Seoul 143-701, Korea E-mail :
[email protected] Sang Mok Choo School of Mathematics and Physics Ulsan University Ulsan 680-749, Korea E-mail :
[email protected] Sang Kwon Chung Department of Mathematics Education Seoul National University Seoul 151-742, Korea E-mail :
[email protected]
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